Asymptotically Accurate Analytical Studies of Elastic Couplings in Anisotropic Beams

Abstract

Elastic coupling is a unique phenomenon exhibited by anisotropic beams. Due to this elastic coupling, different modes of beam deformation, such as extension, bending, and twisting, interact with each other. Modern structures are utilizing this elastic coupling to control them passively. The rotor blade of a wind turbine is one of the most well-known examples, where the bend-twist coupling is being implemented to passively control the angle of attack according to the wind load. This phenomenon of elastic coupling is mainly studied in laminated composite beams, which are generally anisotropic and inhomogeneous. There is a lack of research on elastic coupling in anisotropic-homogeneous beams. In addition, the work that is currently available is based on specific assumptions or has been solved for simplified loading cases. This thesis work investigates the elastic coupling within both anisotropic-homogeneous and anisotropic-inhomogeneous beams. The Variational Asymptotic Method (VAM) has been employed as a mathematical tool, facilitating the simplification of the beam problem. It systematically decomposes the 3D elasticity beam problem into a 2D linear cross-sectional analysis and a 1D non-linear analysis along the beam length. VAM employs the small parameters associated with the beam problem to perform this decomposition, avoiding ad-hoc assumptions. These small parameters are utilized to order the strain energy terms. The procedure begins by considering the dominant terms f irst, then systematically includes lesser dominant terms in higher-order solutions. The analysis of anisotropic-homogeneous beams has been carried out by considering a prismatic beam with solid elliptical cross-section. The study is divided into two parts based on solution characteristics. The first part addresses orthotropic beams, offering solutions for both the Classical and Timoshenko-like beam models. The second part extends the analysis to monoclinic and complete anisotropic beams, providing solutions exclusively for the Classical beam model. For both cases, closed-form expressions for 1D strain measures and displacement fields have been derived, facilitating the recovery of 3D displacement, stress, and strain fields. Notably, it is observed that beams with material anisotropy up to orthotropy do not exhibit elastic coupling phenomena; this phenomenon is first observed in monoclinic material beams and subsequently in complete anisotropic material beams. Furthermore, it is noted that even in complete anisotropic-homogeneous material beams, a fully elastically coupled system is not achieved; instead, only bend-twist coupling is observed. Additionally, the analysis reveals a violation of the plane stress condition in all coupled cases. To validate the results, comparisons have been made with Finite Element Analysis (FEA) and existing literature results, demonstrating a high level of agreement. The analysis of anisotropic-inhomogeneous beam has been carried out using a laminated composite strip-like beam. These laminated composite structures provide the most feasible way to model this type of beam. This analysis is divided into two parts. The f irst part deals with the hygrothermal instabilities of these structures. Hygrothermal stability conditions have been derived using Classical Laminated Plate Theory (CLPT). These conditions have been used to propose the generalized hygrothermally stable stacking sequences with different modes of elastic coupling. Furthermore, these stacking sequences have been optimized to achieve maximum coupling response. The optimized results are compared with conventional numerically optimized results. Additionally, both results are checked for robustness against small perturbations in the optimized results. The comparison shows that the proposed hygrothermally stable stacking sequence provides better results as the number of plies increases. Both stacking sequences show almost similar error distribution in the sensitivity analysis. The second part involves a mathematical analysis of these beams using VAM. Here, nonlinear kinematics for the strip-like beam are presented. The 2D shell membrane and curvature terms are derived from the 3D strain field, enabling the expression of 2D shell parameters in terms of 1D beam parameters. These newly defined 2D shell parameters are utilized to compute the 2D strain energy density functional. The zeroth-order approximate solution is obtained by minimizing the strain energy corrected up to O(Eε2) through the variational principle. This process ultimately yields the linear constitutive relation governing the linear coupling behavior of these beams. To capture nonlinear coupling behavior, the first-order approximate solution is employed. The hygrothermal stability of these structures is verified through FE simulations, using previously optimized hygrothermally stable stacking sequences. The simulation results confirm hygrothermal stability and comparison with FEA results shows a close agreement in the coupling results.